I'd have to check but it was September/October last year. I suspect they'll just argue it was "because of COVID".
What infuriated me was the web portal where they asked me to upload evidence etc had an audit trail and I could see that I'd upload a document on x day and I could see that it had been read but not processed. I had to chase a few times for an update as it was stuck on their end (forget the status message but something along the lines of "awaiting agent update").
I'd 100% complain.
But it depends if you can be bothered and how much you were out of pocket due to their delays. Unless you went to some overseas lender, they will be regulated by the FCA, and there is a strict complaints proceedure they have to follow, EG, acknowledge your complaint within 5 working days, then give you a response within 8 weeks. If you are not happy with their response then you have the right to go to the ombudsman.
Mortgage lenders generally take these things seriously because they are quite hefty fines if they don't. I am not 100% saying they will give you a pay out, but if it were me I'd give it a go.
EDIT: The other thing to consider is, every complaint that gets escalated to the ombudsman, costs the lender £500 a time REGARDLESS on whether it is upheld or not, there are strange rules but something like the first ten are free, then £500 for X amount, then it goes up. So its in teh best interested of the lender to sort it out themselves.
Also, as funny as it sounds, mortgage lenders almost like have complaints registered, because if the lender has very few, the FCA can get a bit funny and look into it as they may be concerned the lender isnt correctly logging or recording their complaints, again, something they have to do under the regulation.
Just say due to the delays that were not your fault, you have been put in financial detriment.